[1]
|
L. Ambrosio, N. Gigli and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2008.
|
[2]
|
H. Bae and R. Granero-Belinchón, Global existence and exponential decay to equilibrium for DLSS-Type equations, J. Dyn. Diff. Equat., (2020).
doi: 10.1007/s10884-020-09852-5.
|
[3]
|
J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307.
doi: 10.1088/0953-8984/17/9/002.
|
[4]
|
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqs., 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y.
|
[5]
|
A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697.
|
[6]
|
P. M. Bleher, J. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923–942.
doi: 10.1002/cpa.3160470702.
|
[7]
|
C. Bordenave, P. Germain and T. Trogdon, An extension of the Derrida–Lebowitz–Speer–Spohn equation, J. Phys. A: Math. Theor., 48 (2015), 485205.
doi: 10.1088/1751-8113/48/48/485205.
|
[8]
|
M. Bukal, E. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numerische Mathematik, 127 (2014), 365-396.
doi: 10.1007/s00211-013-0588-7.
|
[9]
|
M. Bukal, A. Jüngel and D. Matthes, A multidimensional nonlinear sixth-order quantum diffusion equation, Annales de l'IHP Analyse Non Linéaire, 30 (2013), 337–365.
doi: 10.1016/j.anihpc.2012.08.003.
|
[10]
|
M. Burger, L. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM Journal on Imaging Sciences, 2 (2009), 1129-1167.
doi: 10.1137/080728548.
|
[11]
|
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.
doi: 10.1063/1.1744102.
|
[12]
|
J. A. Carrillo, J. Dolbeault, I. Gentil and A. Jüngel, Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1027-1050.
doi: 10.3934/dcdsb.2006.6.1027.
|
[13]
|
J. A. Carrillo, A. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation, Discrete Contin. Dyn. Syst. B, 3 (2003), 1-20.
doi: 10.3934/dcdsb.2003.3.1.
|
[14]
|
J. A. Carrillo and G. Toscani, Long-Time Asymptotics for Strong Solutions of the Thin Film Equation, Commun. Math. Phys., 225 (2002), 551-571.
doi: 10.1007/s002200100591.
|
[15]
|
X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.
doi: 10.1007/s10440-013-9858-8.
|
[16]
|
P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley and S. -M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E, 47 (1993), 4169-4181.
doi: 10.1103/PhysRevE.47.4169.
|
[17]
|
R. Dal Passo, H. Garcke and G. Grün, On a fourth order degenerate parabolic equation: Global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
doi: 10.1137/S0036141096306170.
|
[18]
|
P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.
doi: 10.1007/s10955-004-8823-3.
|
[19]
|
B. Derrida, J. L. Lebowitz, E. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A: Math. Gen., 24 (1991), 4805-4834.
doi: 10.1088/0305-4470/24/20/015.
|
[20]
|
B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.
doi: 10.3934/dcdsb.2010.14.935.
|
[21]
|
D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman & Hall/CRC Numerical Analysis and Scientific Computing. CRC Press, Boca Raton, FL, 2011.
|
[22]
|
J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, Comm. Partial Differential Equations, 38 (2013), 2004-2047.
doi: 10.1080/03605302.2013.823548.
|
[23]
|
L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. PDEs, 13 (2001), 377-403.
doi: 10.1007/s005260000077.
|
[24]
|
U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.
doi: 10.1007/s00205-008-0186-5.
|
[25]
|
A. E. Hosoi and L. Mahadevan, Peeling, healing and bursting in a lubricated elastic sheet, Phys. Rev. Lett., 93 (2004), 137802.
doi: 10.1103/PhysRevLett.93.137802.
|
[26]
|
C. Josserand, Y. Pomeau and S. Rica, Self-similar singularities in the kinetics of condensation, J. of Low Temp. Physics, 145 (2006), 231-265.
doi: 10.1007/s10909-006-9232-6.
|
[27]
|
A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.
doi: 10.1088/0951-7715/19/3/006.
|
[28]
|
A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.
doi: 10.1137/060676878.
|
[29]
|
A. Jüngel and J. -P. Milišić, A sixth-order nonlinear parabolic equation for quantum systems, SIAM J. Math. Anal., 41 (2009), 1472-1490.
doi: 10.1137/080739021.
|
[30]
|
A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal., 32 (2000), 760-777.
doi: 10.1137/S0036141099360269.
|
[31]
|
A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation, SIAM J. Num. Anal., 39 (2001), 385-406.
doi: 10.1137/S0036142900369362.
|
[32]
|
A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete Cont. Dyn. Sys. B, 8 (2007), 861–877.
doi: 10.3934/dcdsb.2007.8.861.
|
[33]
|
J. R. King, The isolation oxidation of silicon: The reaction-controlled case, SIAM J. Appl. Math., 49 (1989), 1064-1080.
doi: 10.1137/0149064.
|
[34]
|
J. R. Lister, G. G. Peng and J. A. Neufeld, Spread of a viscous fluid beneath an elastic sheet, Phys. Rev. Lett., 111 (2013).
|
[35]
|
J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.
doi: 10.1088/0951-7715/29/7/1992.
|
[36]
|
D. Matthes and H. Osberger, A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation, Foundations of Computational Mathematics, 17 (2017), 73-126.
doi: 10.1007/s10208-015-9284-6.
|
[37]
|
T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462.
doi: 10.1137/S003614459529284X.
|
[38]
|
A. Novick-Cohen and A. Shishkov, The thin film equation with backwards second order diffusion, Interfaces and Free Boundaries, 12 (2010), 463-496.
doi: 10.4171/IFB/242.
|
[39]
|
A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.
doi: 10.1103/RevModPhys.69.931.
|
[40]
|
A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, CA, 1951.
|
[41]
|
T. P. Witelski, A. J. Bernoff and A. L. Bertozzi, Blowup and dissipation in a critical case unstable thin film equation, Euro. Jnl. of Applied Mathematics, 15 (2004), 223-256.
doi: 10.1017/S0956792504005418.
|